Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability: Volume 2 Contributions to Probability Theory [Reprint 2020 ed.] 9780520350670

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PROCEEDINGS THIRD BERKELEY

OF

THE

SYMPOSIUM

VOLUME I I

PROCEEDINGS of the THIRD BERKELEY SYMPOSIUM ON MATHEMATICAL STATISTICS AND PROBABILITY Held at the Statistical Laboratory University of California December 26-31, 1954 July and August, 1955

V O L U M E

II

CONTRIBUTIONS TO PROBABILITY T H E O R Y

EDITED BY J E R Z Y

NEYMAN

UNIVERSITY OF CALIFORNIA BERKELEY AND LOS ANGELES

1956

PRESS

UNIVERSITY OF CALIFORNIA PRESS B E R K E L E Y A N D LOS A N G E L E S CALIFORNIA

CAMBRIDGE UNIVERSITY PRESS LONDON, ENGLAND

C O P Y R I G H T , 1 9 5 6 , BY T H E REGENTS OF T H E UNIVERSITY OF CALIFORNIA

T h e United States Government and its officers, agents, and employees, acting within the scope of their duties, may reproduce, publish, and use this material in whole or in part for governmental purposes without payment of royalties thereon or therefor. T h e publication or republication by the government either separately or in a public document of any material in which copyright subsists shall not be taken to cause any abridgment or annulment of the copyright or to authorize any use or appropriation of such copyright material without the consent of the copyright proprietor.

L I B R A R Y O F CONGRESS C A T A L O G C A R D N U M B E R : 4 9 - 8 1 8 9

P R I N T E D IN T H E UNITED STATES OF AMERICA

CONTENTS OF PROCEEDINGS, VOLUMES I, III, IV, AND V Volume I—Statistics JOSEPH BERKSON, Estimation by least squares and by maximum likelihood. Z. W. BIRNBAUM, On the use of the Mann-Whitney statistic. HERMAN CHERNOFF and HERMAN RUBIN, The estimation of the location of a discontinuity in density. ARYEH DVORETZKY, On stochastic approximation. SYLVAIN EHRENFELD, Complete class theorems in experimental design. G. ELFVING, Selection of nonrepeatable observations for estimation. ULF GRENANDER and MURRAY ROSENBLATT, Some problems in estimating the spectrum of a time series. J. L. HODGES, JR. and E. L. LEHMANN, Two approximations to the Robbins-Monro process. WASSILY HOEFFDING, The role of assumptions in statistical decisions. SAMUEL KARLIN, Decision theory for Pólya type distributions. Case of two actions, I. L. LE CAM, On the asymptotic theory of estimation and testing hypotheses. HERBERT ROBBINS, An empirical Bayes approach to statistics. MURRAY ROSENBLATT, Some regression problems in time series analysis. CHARLES STEIN, Efficient nonparametric testing and estimation. CHARLES STEIN, Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. B. L. VAN DER WAERDEN, The computation of the X-distribution.

Volume III—Astronomy and Physics O. J. EGGEN, The relationship between the color and the luminosity of stars near the sun. J. L. GREENSTEIN, The spectra and other properties of stars lying below the normal main sequence. H. L. JOHNSON, Photoelectric studies of stellar magnitudes and colors. G. E. KRON, Evidence for sequences in the color-luminosity relationship for the M-dwarfs. G. C. McVITTIE, Galaxies, statistics and relativity. JERZY NEYMAN, ELIZABETH SCOTT and C. D. SHANE, Statistics of images of galaxies with particular reference to clustering. BENGT STRÓMGREN, The Hertzsprung-Russell diagram. F. ZWICKY, Statistics of clusters of galaxies. ANDRÉ BLANC-LAPIERRE and ALBERT TORTRAT, Statistical mechanics and probability theory. M. KAC, Foundations of kinetic theory. J. KAMPÉ DE FÉRIET, Random solutions of partial differential equations. E. W. MONTROLL, Theory of the vibration of simple cubic lattices with nearest neighbor interactions. NORBERT WIENER, Nonlinear prediction and dynamics.

Volume IV—Biology and Problems of Health JAMES CROW and MOTOO KIMURA, Some genetic problems in natural populations. E. R. DEMPSTER, Some genetic problems in controlled populations. JERZY NEYMAN, THOMAS PARK and ELIZABETH SCOTT, Struggle for existence. The Tribolium Model: biological and statistical aspects. M. S. BARTLETT, Deterministic and stochastic models for recurrent epidemics. A. T. BHARUCHAREID, On the stochastic theory of epidemics. C. L. CHIANG, J. L. HODGES, JR. and J. YERUSHALMY, Statistical problems in medical diagnoses. JEROME CORNFIELD, A statistical problem arising from retrospective studies. D. G. KENDALL, Deterministic and stochastic epidemics in closed populations. W. F. TAYLOR, Problems in contagion.

Volume V—Econometrics, Industrial Research, and Psychometry K. J. ARROW and LEONID HURWICZ, Reduction of constrained maxima to saddle-point problems. E. W. BARANKIN, Toward an obj'ectivistic theory of probability. C. W. CHURCHMAN, Problems of value measurement for a theory of induction and decisions. PATRICK SUPPES, The role of subjective probability and utility in decision-making. A. H. BOWKER, Continuous sampling plans. CUTHBERT DANIEL, Fractional replication in industrial research. MILTON SOB EL, Sequential procedures for selecting the best exponential population. T. W. ANDERSON and HERMAN RUBIN, Statistical inference in factor analysis. FREDERICK MOSTELLER, Stochastic learning models. HERBERT SOLOMON, Probability and statistics in psychometric research: Item analysis and classification techniques.

ACKNOWLEDGMENT Most of the papers published in this volume were delivered at the sessions of the Symposium held in the summer of 1955. These sessions were organized in cooperation with the Institute of Mathematical Statistics Summer Institute, 1955, Professor David Blackwell, Chairman.

PREFACE on Mathematical Statistics and Probability was held in two parts, one from December 26 to 31, 1954, emphasizing applications, and the other, in July and August, 1955, emphasizing theory. The Symposium was thus divided because, on the one hand, it was thought desirable to provide an opportunity for contacts between American and Foreign scholars who could come to Berkeley in the summer, but not in the winter, and because, on the other hand, the 121st Annual Meeting of the American Association for the Advancement of Science held in Berkeley in December, 1954, provided an opportunity for joint sessions on the various fields of applications with its many member societies. With the help of Dr. Raymond L. Taylor, of the AAAS, nine cosponsored sessions of the Symposium were organized. Two of these were given to problems of astronomy and one each to biology, medicine and public health, statistical mechanics, industrial research, psychometry, philosophy of probability, and to statistics proper. The importance of the second part of the Symposium, which emphasized theory, was increased by the decision of the Council of the Institute of Mathematical Statistics to hold its first Summer Institute in Berkeley and to hold this Institute "in conjunction with the Third Berkeley Symposium"; all members of the IMS Summer Institute were invited to participate in the Symposium and the two enterprises were conducted in parallel. In particular, the cooperation of Professor David Blackwell, Chairman of the IMS Summer Institute, made it possible to ensure that representatives of, all the major centers of statistical research in this country be invited. As will be seen from the lists of contents of the Proceedings, the response was good, although various circumstances, including the concurrent Rio de Janeiro meeting of the International Statistical Institute, prevented some of the prospective participants from attending the Berkeley meetings. Two months were allotted to the second part of the Symposium in order to provide an opportunity not only for formal presentation of papers, but also for informal contacts among the participants. To facilitate such personal associations, after three weeks of intensive lectures and discussions, a trip was made to the Sierra. There, animated discussions of stochastic processes and of decision functions were interspersed with expressions of delight at the beauty of Yosemite Valley, Emerald Bay, and Feather River Canyon. After this vacation there was another period of intensive lecturing. Although much effort was expended to arrange lectures and personal contacts, the primary concern of the Statistical Laboratory and of the Department of Statistics was with the Proceedings. Because of the participation of the AAAS, the amount of material submitted for publication was estimated to be equivalent to thirteen hundred printed pages, roughly twice the length of the Proceedings of the Second Berkeley Symposium. This presented a most embarrassing problem. That it was finally solved is largely the result of the most effective support and advice of Dr. John Curtiss, Executive Director of the American Mathematical Society. His organizational talent and friendly help are greatly appreciated. Special thanks are due Mr. August Fruge, the Manager of the Publishing Department of the University of California Press, and also his staff, who undertook the difficult and costly publication in the best spirit of cooperation with, and of service to, the scholarly community. T H E T H I R D B E R K E L E Y SYMPOSIUM

vii

viii

PREFACE

Since a single thirteen-hundred-page volume would have been difficult to handle and, for the majority of scholars, too expensive to buy, it was decided to issue the Proceedings in five relatively small volumes, each given to a specialized and, so far as possible, unified cycle of ideas. A list of contents of the other four volumes of the Proceedings will be found preceding this preface. The initial steps in the organization of the Symposium were based on a grant obtained from the University of California through the good offices of Professor Clark Kerr, Chancellor of the Berkeley campus of the University of California, to whom sincere thanks are due. This grant was followed by an appropriation from the Editorial Committee of the University of California, which provided the nucleus of the fund eventually collected for the publication of the Proceedings. This action of the Editorial Committee is gratefully acknowledged. For further effective support of the Symposium thanks must be given the National Science Foundation, the United States Air Force Research and Development Command, the United States Army Office of Ordnance Research, and the United States Navy Office of Naval Research. It is hoped that the material in the present Proceedings and, particularly, the scientific developments stimulated by the Symposium, will be sufficiently important to prove that the money received from these organizations was well spent. The success of the Symposium was, in large part, made possible by the generous and effective support of a number of scholarly societies. Sessions of the first part of the Symposium were sponsored by the American Physical Society; the American Statistical Association; the Astronomical Society of the Pacific; the Biometric Society, Western North American Region; the Ecological Society of America; the Institute of Mathematical Statistics; the Philosophy of Science Association; and the Western Psychological Society. The American Mathematical Society sponsored the second part of the Symposium, delegating for organizational work two of its most distinguished members, Professor J. L. Doob and Professor G. Polya, whose advice and cooperation were most helpful. The 1955 Summer Institute of the Institute of Mathematical Statistics was held in conjunction with the Symposium; the IMS also held its Western Regional Meeting in Berkeley in July. All papers published in these Proceedings were written at the invitation of the Statistical Laboratory, acting either on its own initiative or at the suggestion of the cooperating groups, and the Laboratory is, therefore, responsible for the selection of the authors, a responsibility that does not extend to the contents of the papers. The editorial work on the manuscripts submitted was limited to satisfying the requirements of the University of California Press regarding the external form of the material submitted, the numbering of formulas, etc., and to correcting obvious errors in transcription. Most of this was done by the staff of the Laboratory, in particular, Miss Catherine FitzGibbon, Mrs. Jeanne Lovasich, Mrs. Kathleen Wehner, and my colleague, Dr. Elizabeth L. Scott, who supervised some of the work. Occasionally, manuscripts were read by other participants in the Symposium particularly interested in them, and the authors were offered suggestions. However, in no case was there any pressure on the authors to introduce any material change into their work. JERZY NEYMAN

Director, Statistical Laboratory Chairman, Department of Statistics

CONTENTS D A V I D BLACKWELL—On

a Class of Probability Spaces

1

S. BOCHNER—Stationarity, Boundedness, Almost Periodicity of RandomValued Functions

7

K. L. CHUNG—Foundations of the Theory of Continuous Parameter Markov Chains

29

A. H.

.

41

J. L. DOOB—Probability Methods Applied to the First Boundary Value Problem

49

FORTET—Random Distributions with an Application to Telephone Engineering

81

COPELAND,

SR.—Probabilities, Observations and Predictions

.

ROBERT M .

J. M. HAMMERSLEY—The Zeros of a Random Polynomial T. E. HARRIS—The Existence of Stationary Measures for Certain Markov Processes KIYOSI

ITO—Isotropic Random Current

Special Problem of Brownian Motion, and a General Theory of Gaussian Random Functions

89 113 125

P A U L L£VY—A

MICHEL

LofevE—Ranking Limit Problem

133 177

LUKACS—Characterization of Populations by Properties of Suitable Statistics

195

MENGER—Random Variables from the Point of View of a General Theory of Variables

215

MOURIER—¿-Random Elements and L*-Random Elements in Banach Spaces

231

R. SALEM and A. ZYGMUND—A Note on Random Trigonometric Polynomials

243

EUGENE

KARL

EDITH

ON A CLASS OF PROBABILITY SPACES DAVID BLACKWELL UNIVERSITY OF CALIFORNIA, BERKELEY

1. Introduction Kolmogorov's model for probability theory [10], in which the basic concept is that of a probability measure P on a Borel field B of subsets of a space fi, is by now almost universally considered by workers in probability and statistics to be the appropriate one. In 1948, however, three somewhat disturbing examples were published by Dieudonne [2], Andersen and Jessen [1], and Doob [3] and Jessen [9], as follows. A. (Dieudonne). There exist a pair (fi, a probability measure P on jS, and a Borel subfield rf c B for which there is no function Q(u, E) defined for all to £ i), E 6 B with the following properties: Q is for fixed E an /^-measurable function of u, forfixed and for every A £ /Q, E £ B> we have (1)

J Q (w, E) dP (w) =P (A

r\E).

B. (Andersen and Jessen). There exist a sequence of pairs (fi„, Bn) and a function P defined for all sets of u where / J n consists of all subsets of the infinite product space fli X fl« X • • • in the Borel field determined by sets of the form Bi X • • • X Bn X Qn+i X iin+2 X • • •, Bi £ Bi, i = 1, • • •, n, such that P is countably additive on each / ¡ n but not on u / J n . C. (Doob, Jessen). There exist a pair (0, £ ) , a probability measure P on B> and two real-valued ^ - m e a s u r a b l e functions/, g on 0 such that (2)

P { « : / € F , geG\

=P{u:

f£F\P{a,:

g € G}

holds for every two linear Borel sets F, G but not for every two linear sets F, G for which the three probabilities in (2) are defined. In each case i2 is the unit interval, Q is the Borel field determined by the Borel sets and one or more sets of outer Lebesgue measure 1 and inner Lebesgue measure 0, and P consists of a suitable extension of Lebesgue measure to The fact that A, B,C cannot happen if fl is a Borel set in a Euclidean space and £ consists of the Borel subsets of C2 is known. For A, the proof was given by Doob [4], for B by Kolmogorov [10], and for C by Hartman [7]. To the extent that A, B,C violate one's intuitive concept of probability, they suggest that the Kolmogorov model is too general, and that a more restricted concept, in which A, B,C cannot happen, is worth considering. In their book [5], Gnedenko and Kolmogorov propose a more restricted concept, that of a perfect probability space, which is a triple (0, Bi P) such that for any real-valued ^-measurable function / and any linear set A for which {w:/(o>) G A\ d B, there is a Borel set B c A such that (3)

P { w : / ( « ) € £ } =P{o>:/(«)€

This investigation was supported (in part) by a research grant from the National Institutes of Health, Public Health Service. I

2

THIRD BERKELEY SYMPOSIUM: BLACKWELL

As noted by Doob [4] (see appendix in [5]) in perfect spaces A, C cannot happen, and it then follows from a theorem of Ionescu Tulcea [8] that B cannot happen in perfect spaces. The concept introduced by the writer here is that of a Lusin space, which is a pair (0, JS) such that (a) fe is separable, that is, there is a sequence \Bn) of elements of & such that fe is the smallest Borel field containing all Bn, and (b) the range of every realvalued ^-measurable function / on Q is an analytic set, that is, a set which is the continuous image of the set of irrational numbers. The concept of Lusin space is more restricted than that of perfect space in the sense that if (fi, j2) is a Lusin space and P is any probability measure on Q, then (0, Q, P) is perfect. It is shown below that for Lusin spaces none of A, B, C can occur. The primary property of Lusin spaces which ensures this regularity, and which fails for the example of A, B, C mentioned above, is that the only events whose occurrence or nonoccurrence is determined by specifying which events in a sequence E\, ¿£2,' ' ' occur are the events in the Borel field determined by the sequence {£„}. This property permits the identification of the concepts, for real-valued ^-measurable functions/, g, "f is a function of g" and "/is a Baire function of g," the nonequivalence of which in general is a technical nuisance to say the least. 2. Preliminaries In this section we list some definitions and some known properties of analytic sets to be used in later sections. If M is a metric space, the sets in the smallest Borel field containing all open sets will be called the Borel sets of M. A Borel field Q of subets of a space 0 will be called separable if there is a sequence {Bn} of sets in fe such that fc is the smallest Borel field containing all Bn. Thus if 0 is a separable metric space, the class of Borel sets is a separable Borel field, though not conversely. If H is a separable Borel field of subsets of fi and {Bn\ is a sequence determining j8, the sets of the form n C„, where each C„ is either Bn or 12 — Bn, are called the atoms of Q. Any two nonidentical atoms are disjoint and every set in & is a union of atoms, so that the class of atoms of & is independent of the particular sequence {2?nj. A metric space A will be called analytic if A is the continuous image of the set of irrational numbers. We shall use the following properties of analytic sets, due to Lusin [11], I. If {A n \ is a sequence of analytic sets in a metric space M, then u An, n An if nonempty, the product space Ai X A2, and the infinite product space Ax X i j X ' " are analytic sets. II. If A is analytic, so is every Borel subset of A. III. Every Borel set of Euclidean »-space is analytic. IV. If A, B are disjoint analytic subsets of a metric space M, there is a Borel set D of M such that D s A and D is disjoint from B. V. If / is a Borel-measurable mapping of an analytic set A into a separable metric space M, that is, the inverse image of any open set in If is a Borel set of A, then f A, the range of/, is an analytic set. We shall also use the following property pointed out to the writer by A. P. Morse. VI. If P is a probability measure on the Borel sets of a metric space M and A is an analytic subset of M then for every e > 0 there is a compact C inside A with P{C) > jtt — e, where n = min P{B) as B varies over all Borel sets of M which contain A.

ON A CLASS OF PROBABILITY SPACES

3

3. Lusin spaces and analytic sets The main content of theorems 1 and 2 is that, apart from the unessential difference that the atoms of a Lusin space need not be points, Lusin spaces are identical with pairs (Q, /}) where Si is analytic and >3 is the class of Borel sets of £2. THEOREM 1 . 7 / 0 is analytic and fc is the class of Borel sets of U, then (Q, j§) is a Lusin space. PROOF. Separability of fc follows from the separability of 12, and that the range of every ^-measurable real-valued / is analytic is the special case of V with M the real line. THEOREM 2 . If ( 0 , /£) is a Lusin space whose atoms are points and {£„} is any sequence determining fc then there is a metric on i2 with respect to which 0 is an analytic set, fe consists of the Borel sets of 0, and every En is both open and closed. PROOF.

Say {£„} determines fc and let/(w) =

^ ^ E„(CO)/3N

where en is the charac-

n

teristic function of En. Then / is a 1-1 ^-measurable map of Q, onto an analytic subset A of the line. Letd(o>i, a>2) = a)2), where k is the smallest «for which e„(o)i) ^ en(oa). Then / is bicontinuous between 0 and A, and every E„ is open and closed, since any point not in En has distance at least l/n from En. Finally, to identify Q with the class ,2J of images of Borel sets of A under/-1, the ^g-measurability of / implies JS 3 2>, and we need only show En 6 3> t o conclude 2> 3 J@- Since E„ is the image under/ -1 of the set of numbers in A whose wth ternary digit is 1, the proof is complete. 4. Set theoretic properties of Lusin spaces THEOREM 3 . If (Q, J£) is a Lusin space, Q is a separable Borel field of Q-sets and A is a union of atoms of Q, then A £ Q-

PROOF. Say {CN} determines Q and let/(w) = ^

6

B

c„(A>)/3n. Then / maps (J-atoms n into points, and different £-atoms into different points. Then/^4 and/(i2 — A) are disjoint analytic linear sets, so that from property IV there is a linear Borel set D such that D D f A and D is disjoint from /(Q — A). Consequently f~lD = A, so that, since / is (^-measurable, A £ Q. COROLLARY 1. If ( 0 , /¡) is a Lusin space, two separable Borel fields of jS-sets with the same atoms are identical. COROLLARY 2 . Let (I2, /¡) be a Lusin space and let f map il onto an arbitrary space Z. If there is a separable Borel field Q c £ whose atoms are the sets /_1(z), z £ Z, then Q is identical with the class of all sets in £ of the form f~xD, D c Z, and (Z, 2b) is a Lusin space, where consists of all D £ 2> for which f~xD £ Q. PROOF. The mapping / - 1 is a 1-1 mapping between the points z £ Z and the atoms of Q. Thus every C € Q has the form / - 1 D for some D c Z. Conversely if A = f~lD for some D c Z, A is a union of atoms of Q, so that, from the theorem, A £ £ implies A 6 Q. Thus D 6 3> if and only if f~lD £ Q. It follows that if {C„} generates Q, then {fCn} generates 2>, so that is separable. Finally, if h is any real-valued ^-measurable function on Z, then hf is a ^-measurable function on 0 whose range is the same as the range of h. Since (0, jS) is a Lusin space, this range is analytic and the proof is complete. COROLLARY 3 . Let ( 0 , /¡) be a Lusin space, let f map 0 onto an arbitrary space Z, and denote by £ the Borel field of all Z-sets S for which f^S € fe. For any separable 3> c £>

4

THIRD BERKELEY SYMPOSIUM: BLACKWELL

(Z, 2>) is a Lusin space. If in addition every 5 € £ is a union of atoms of 2b, then = so that (Z, g) is a Lusin space. PROOF. The first conclusion of the corollary follows immediately from the definition of a Lusin space. The second conclusion follows from the first and theorem 3. COROLLARY 4 . If (fi, jB) is a Lusin space and f is a /¡-measurable function from fi into a separable metric space M, then for every set A c M for whichf~lA £ B there is a Borel set B of M such that f~lB = f~lA. PROOF. The class Q of all sets of the form/ _ 1 £, where B is a Borel set of M, is a separable Borel subfield of B. Every set f_1A is a union of atoms of Q, and thus is in Q if it is in B Thus if f~lA 6 jg, there is a Borel set B of M for which f~xB = f'xA. Theorem 3 identifies, for Lusin spaces, the concepts "an event A depends only on events in Q " and "A £ Q . " The following theorem extends this to functions. THEOREM 4 . Let (£2, B) be a Lusin space, let f , g be jS-measur able functions from fi into separable metric spaces Y, Z, and denote by BA&a) the class of all sets of the form f^S (g-^T) where S(T) is a Borel set in Y(Z). (a) If there is a function from Y into Z such that f = g, then g is measurable with respect to the Borel field of f -sets, that is, Bq c Bi(b) If Ba c jSf, then there is a B orel-measurable function from Y into Z such that PROOF, (a) Separability of F implies separability of j£/. Every set in is a union of atoms of H¡, so that (a) follows from theorem 3. The hypothesis that (0, ff) is a Lusin space is not necessary for (b); the proof by Doob [4] for F, Z Euclidean spaces extends easily to arbitrary separable metric spaces F, Z. 5. Conditional probability distributions, Kolmogorov extension THEOREM 5 . Let ( f l , /£) be a Lusin space, let P be a probability measure on Q, and let ^ be a separable Borel subfield of Q. There is a real-valued function Q(a>, B) defined for all w £ Q, B £ j£ such that (a) for fixed B € Q is an rf-measurable function of (b) for fixed w, Q is a probability distribution on Q,

(c) for every A € A, B 6 B,

Jf

Q(u, B)dP = P(A n B), and A (d) there is a set N € A with P(N) = 0 such that Q{w, A) = 1 for w £ A, ai $ N. PROOF. We may suppose that the atoms of fc are points. Choose Fn 6 B so that {Fn} determines B and a subsequence of {F„} determines / J , and (theorem 2) metrize Q. so that fl is analytic, B consists of the Borel sets, and each Fn is open and closed. Choose C„ compact so that C„ c C„+i, P(Cn) —> 1, and denote by Q the fields determined by Fh Flt • • • and Fh F2, • • •, Ci, C2, • • • respectively. Let Qi(oi, B) be defined so as to satisfy (a) and (c) for B 6 Q- Since Q is countable, there is a set N £ /J with P(N) = 0 such that for w i N, (4)

Qi is additive and nonnegative on Q ,

(5)

Qi (co, ii) = 1 ,

(6)

Qiia.,

for

AZA,

as

»-»co.

and (7)

A^Q

ON A CLASS OF PROBABILITY SPACES

S

T h e n Qi is countably additive on for if {H„} is a sequence of disjoint sets of ¿p with u Hn = i2, for every n, since C„ is compact and the Hn are open and closed, there is an M

M such t h a t U H 3 C„. Finite additivity of Q\ on G yields, for co £ N, Qi(co, C„) ^ 1 M co co Hi) g ^ (^(co, Hi). Letting » - > oo and using (7) yields ^ Q^oi, Hi) ^ 1. i i i Additivity yields the reverse inequality, so t h a t , for co £ N , Qi is countably additive on ¿5?. For co £ N, we define Q(co, 5 ) as the (unique) countably additive extension of Qi from to jg. For co £ N, we define Q(u, = P(B). T h e n (b) holds, and the class of sets B for which (a) and (c) hold is a monotone class containing ¿p, so coincides with Q [6]. To verify (d) let A £ / J , oj £ A, co £ N, and denote by I the /J-a.torn containing co. T h e n I c A and, since a subsequence of {F n j determines / ] , there is a sequence / „ £ 4Z for which nJn = I. F r o m (2),Q(w, / „ ) = 1 for all n, so t h a t Q(u, I ) = 1. T h i s completes the proof. THEOREM 6. Let {(0„, /gn)} be a sequence of Lusin spaces, let Q be the infinite product space fiiX 0 2 X ' • •, and let /Jn be the Borel field determined by all sets Ai X • • • X An X fi„+i X X ' • ' , Ai £ jSi. A function P defined on u /JH which is a probability measure on each is countably additive on u /¡n. PROOF. L e t An be a decreasing sequence of sets in U with P(An) —* 28 > 0. W e m u s t show t h a t n An is not empty. We m a y suppose t h a t the atoms of are points and metrize Qn so t h a t it becomes analytic and jgn consists of the Borel sets of i2„. We m a y also suppose t h a t An £ /}„. From property V I there is a set Dn £ /Jn such t h a t Dn c An, P(Dn) > P(An) - 5/2", and Dn = CnX 0„+i X O n + 2 X • where C„ is a compact subset of 0X X • • • X 0». Since P(DX n • • • n DN) > P{A-i n • • n N — S ^^ 2~n > S > 0, A n • n DN is nonempty for each N. If u>n — {xn\, •"") € i Di n • n Dn , we have (xm, • • ' , xNk) £ Ck for all N ^ k, so that there is a subsequence of coiv which converges coordinatewise to a point co* — (£*, x*,- • •), and («*,• • •, xt) £ Ck for each k. T h u s co* £ Dk c Ak for all k and the proof is complete.

6. Independence, perfection THEOREM 7. If (fi, Jg) is a Lusin space, P is any probability measure on /g, and f , g are any two /¡-measurable functions from 0 into separable metric spaces X, Y such that (8)

P { c o : / £ A , geB}

= P{co:/£

A}P{g£B\

for all Borel sets A, B in X, Y, then (8) holds for all sets A, B in X, Y for which the terms are defined. PROOF. T h e theorem follows immediately from corollary 4 of theorem 3. THEOREM 8. If (Q, /g) is a Lusin space, P is any probability measure on /g andf is any /¡-measurable function from 0 into a separable metric space, then inside any A c M for which f ' 1 A £ jg there is a Borel set B of M with P{f~lB) = Pif^A). PROOF. W e m a y suppose A c R, the range of / . If Q consists of the Borel sets of R, then (R, Q) is a Lusin space and, from corollary 4 of theorem 3, A £ Q. T h e function (C) = P ( / _ 1 C ) is a probability measure on Q, and from property V I there is inside A a union B of compact sets with (B) = (A).

6

THIRD B E R K E L E Y SYMPOSIUM :

BLACKWELL

7. Some unsolved problems Problem 1. If ß is a separable Borel field of subsets of a space 0 such that every separable Borel subfield of ß with the same atoms is identical with ß, is (Î2, ß) a Lusin space? Problem 2. If ß is a separable Borel field of subsets of a space ß such that (0, ß, P) is perfect for every probability measure P on ß, is (0, ß) a Lusin space? Problem 3. Can the exceptional set N be eliminated from (d) of theorem 5? That is, given an analytic set N and a separable Borel subfield /J of the class ß of Borel sets of N, does there exist a function Q(u, B) defined for all co G N and all £ 6 ß which (i) for fixed B is an ^-measurable function of w, (ii) for fixed co is a probability distribution on ß and (iii) for whichQ(co, A) = 1 for all A 6 /J, a € A? REFERENCES [1] E. SPARRE ANDERSEN and B0RGE JESSEN, "On the introduction of measures in infinite product sets," Dattske Vid. Selsk. Mat.-Fys. Medd., Vol. 25, No. 4 (1948), 8 pp. [2] JEAN DIEUDONNÉ, "Sur le théorème de Lebesgue-Nikodym. I I I , " Ann. Univ. Grenoble, Vol. 23 (1948), p p . 2 5 - 5 3 .

[3] J. L. DOOB, "On a problem of Marczewski," Colloquium Mathematicum, Vol. 1 (1948), pp. 216-217. [4] •— -, Stochastic Processes, New York, John Wiley and Sons, 1953. [5] B. V. GNEDENKO and A. N. KOLMOGOROV, Limit Distributions for Sums of Independent Random Variables. Translated by K. L. Chung with an appendix by J. L. Doob, Cambridge, AddisonWesley, 1954. [6] P. L. HALMOS, Measure Theory, New York, D. Van Nostrand, 1950. [7] S. HARTMAN, "Sur deux notions de fonctions indépendantes," Colloquium Mathematicum, Vol. 1 (1948), pp. 19-22. [8] C. T. IONESCU TULCEA, "Measures dans les espaces produits," Atti Accad. Naz. Lincei Rend. Cl. Sei. Fis. Mal. Nat., Vol. 7, Series 8 (1949), pp. 208-211. [9] B0RGE JESSEN, "On two notions of independent functions," Colloquium Mathematicum, Vol. 1 (1948), p p . 2 1 4 - 2 1 5 .

[10] A. N. KOLMOGOROV, Grundbegriffe der Wahrscheinlichkeitsrechnung, Ergebnisse Mathematik, Berlin, Springer, 1933. [11] NICOLAS LUSIN, Leçons sur les Ensembles Analytiques et leurs Applications, Paris, Gauthier-Villars, 1930.

STATIONARITY, BOUNDEDNESS, ALMOST PERIODICITY OF RANDOMVALUED FUNCTIONS S. B O C H N E R PRINCETON UNIVERSITY

1. Introduction T h e present paper is an enlarged version of the previous paper [1], to which we shall refer occasionally, b u t the notation has been changed in places. T h e problem is as follows. Let x{t) on — °° < t < oo ,

and that this is also so for the general operator (2.2.1) provided (2.2.8)

inf

-co 0, uniformly for n Si 1, provided X{T)/T—> 0 as r —* 0. But the latter takes place for almost all values of t, and this completes the proof of the theorem. 4.3. If a complex function £(/) is F-bounded, then so is the function c£(t) for any element c in B. Also, the sum of F-bounded functions is again F-bounded. Hence the first part of the following theorem. THEOREM 4.3.1. For any B, a finite sum (4.3.1)

£ m=l

is V-bounded for any real am and any cm in B.

RANDOM-VALUED FUNCTIONS

21

Also, if x(t) is V-bounded then so is also any smoothing xv(t) as defined by (2.4.7).

The second half follows immediately from the fact the Fourier transform of (-«)#>(-

v) x(u) x(v)

' J — CO J— 00

00 /• oo

dudv\

[

— JI Co /*/—CO 00 , uniformly in — 0 (2.6)

Pim(t)

=P{x{t,u)

= oo | x ( 0 , w) = i } = 0 ,

since ^ pij(t) = 1. Furthermore since ^ pi — 1 we have also P{x(t, a) = oo } = 0 0SJ') with i instantaneous; and a set of measure zero which is contained in the closure of the set of jumps. Each of the open intervals whose union is the (;possibly empty) complement of D(u>) belongs to a certain Si(o>) with i stable, and two adjacent ones belong to different S), x(-, co) is constant and equal to i. PROOF. L e t C(co) be the union of all Si(co) with i instantaneous. B y property (iv), section 4, C(co) C D(u). I f t £ D(w) — C(co) and i is an instantaneous state then there is a neighborhood of t in which *(•, co) ^ i. Hence by theorem D we have as t —»r from one side, the following possibilities with probability one: (i) lim x(t, co) = i j = lim x(t, co) where i and j are stable; ¿ = lim x(t, w) where /|t /j. t f\r ¡ir 1 is stable; (iii)

lim x(t, co) = oo ;

(iv)

x(t, co) has two limiting values i and

t—

00

where i is stable.

T h e set NI of co for which (iv) is true for some T has probability zero, by theorem 1. Because of separability we have X(T, co) = oo in case (iii) if r does not belong to the denumerable set satisfying the conditions of the separability definition; while if it belongs to this set then case (iii) has probability zero by theorem 3 applied to all r in the denumerable set. Now the set E of (r, co) for which x{j, to) = oo is measurable and for each fixed r the co-set {co: (r, co) £ E\ has probability zero by theorem 2 (iii). Hence by Fubini's theorem if co is not in a set N i with P(Ní) = 0 the set of T for which (iii) is true has measure zero. Finally let f¿0 be the set specified in theorem 1. Then for each co 6 Oo the set of r for which either (i) or (ii) is true may be put into an at most 2-1 correspondence with the set of stable intervals of co). Hence it is a denumerable set by theorem 1. W e have thus proved that for every co £ Í20 — Ni — = 0 4 , m[D(co) — C(co)] = 0. Since the possibility (iv) has been excluded, each of the remaining possibilities presents a discontinuity r which is either a jump or a limit point of jumps. T h e other assertions in theorem 3 merely give a résumé of some of the previous results.

36

THIRD BERKELEY SYMPOSIUM: CHUNG

Theorem 3 reduces to a result by Levy (see p. 349 in [4]) if there are no instantaneous states. Note that the value of x(', co) is not uniquely prescribed by separability at certain points, for example, at a point of Si(o>) n Sj(oi), i ^ j. For the adjoined, state °° the set S^tii) may be further specified as follows: r £ Sm(u>) if and only if lim x{t, co) = In f—•T fact, at any other point separability does not prevent us from changing the value of X(T, CO) to one of the finite limiting values of x(t, co) as t —»r. The resulting process remains measurable since changes are made only on a (t, co) set of measure zero. We are now in a position to strengthen theorem D as follows: THEOREM 4. For almost all co, the sample function x(-, co) has the following properties. For every r, as t f r or t J, T we have one of the following possibilities: (i) x(t, co) —> i where i is stable; (ii) x(t, co) has exactly two limiting values i and °° where i is instantaneous; (iii) x(t, co) —> a>. Furthermore, if X(T, CO) = i where i is stable then (i) is true with the same i as t —> r from at least one side; if X(T, CO) = i where i is instantaneous then (ii) is true with the same i as t—*T from at least one side; if x(j, co) = oo then (iii) must be true as t—*r from both sides. PROOF. According to theorem D these are the three possibilities with the state i in (i) and (ii) yet unspecified. Now by property (iv) of section 4 the probability is zero that x(t, co) —> i with i instantaneous, hence in (i) the state i must be stable. By theorem 1 the probability is zero that a stable i is a limiting value of x(t, co) as t —»r from one side without being the limit. Hence in (ii) the i must be instantaneous. The remaining assertions follow from the separability of the process and the remark preceding theorem 4. 6. System theorems In this section we prove several results concerning the optional starting, stopping and splitting of the process. Such theorems have their origin in so-called gambling systems, hence the name "system theorems." They were frequently regarded as obvious and used without comment. Let 4?{x(s, co), j < /}, or 4?{x(s, co), s ^ t], be the Borel field of co-sets generated by the random variables x{s, co) with s < t, or s ^ t. Let a(co) be a nonnegative random variable such that for every t > 0 (6.1)

{ « ( « ) < * } € 4 * { * ( i , « ) . * « J : 00

(6.6)

P (A2) = lim V p {

(r - 1) h g t (co) < rh; A 2 }

CO

= lim V P {

(r-l)Agr(co)

< r / * < r ' (co) ; A2 ( r A ) }

sinceO < t'(co) — t(co) < °° with probability one, and on the set {co:t(co) < rh < t'(co)} the set A 2 becomes A2(rA). Now the set {co: t'(co) > rh] differs from a set in ^{x(s, co), 5 < rh} by a set of probability zero; hence we may use the Markovian property to obtain CO ( 6 . 7 ) P ( A 2 ) = l i m V p { ( f - l ) i g r ( a ) o^TÍ •P{ x (t, co) = t , rA ^ t ^ rfe + a \ x (rh, co) = i } •P{A.z(rh

+ a) \x(rh

+ a,oi)

= »}

00

= l i m P { A i , ( r — Í) h
) }.

Then for every a >

0

and

P{ X (co) > a; A} = P (A) e~q'a .

(6.9)

PROOF. Consider a set of t h e f o r m 00 (i) (9) A0 = U Am Am m= 1

(6.10)

where the A^,1' and are disjoint sets in #{x(s, co),í ^ r(co)} and #{x(s, co), .5 5; r'(co)} respectively. Such sets form a field Qa containing both these Borel fields. We have (6.11)

P { X (co) >a- Ac} = 2 ) P { X > a ; A ^ ; A ? } m = £P(A^)P(AÍ?) m

Putting a = 0 we obtain (6.12)

P(Ao) = ^ P Í a L V í a Í ) . m

Hence we have for every Ao € a; Ao} — P (Ao) e_5»a .

Thus (6.9) is true for every set in Qt¡ and consequently also for every set in Q. COROLLARY 2. Suppose that all r„(co, i), r'n{co, ¿) awd X„(co, t) /or all stable i and n are defined with probability one. Then every finite set of random variables X„„(co, i,),v = 1, • • •, N, such that if ¡i v then ^ i„ or n„ ^ w„, w a se/ of independent random variables. This follows from repeated application of corollary 1. Corollary 2 is stated by Lévy (see p. 349 in [4]) and is essential for much of his work there. Because of the relative recent growth of rigor in the discussion of stochastic processes we permit ourselves the following remarks. Consider, for example, the successive X„(co, i),n ^ 1, for a fixed i. By an abuse of the Markov property it seems obvious that these random variables are independent and identically distributed. It would then be easy, for example, to deduce theorem 1, namely, that their number is finite in every finite ¿-interval. However, we wish to stress the point that such a procedure cannot possibly be justified. In fact, theorem 1 must precede corollary 2 above in the logical order, because the random variables X„(co, i) cannot be defined without theorem 1. Let us suppose for the sake of argument that the latter theorem merely asserted that there is at most a denumerable number of »-intervals in every finite /-interval. It would then be impossible to define the X„(co, i) unless the set of »-intervals were first shown to be well ordered. Perhaps this is one reason why Doob in his 1945 paper [2] restricted himself to processes whose discontinuities are well ordered in time. THEOREM 6. Let a(co) be the least tfor which t G Si(u). Suppose that the initial distribution is such that P{a(co) < °° } = 1. Let A 6 &{x(s, co), s ^ a(co)}, 0 < h < t2 < • • • < tN, ¿1,'"', is, be any states. We have then (6.14)

P{A; x [a (co) +U, co] = í , ,

íúv^N}

= P (A) p u ^ h ) p i ^ i h - ti) • • • p Í!f _ lÍN ( h - t s - J .

CONTINUOUS PARAMETER MARKOV CHAINS PROOF. L e t S+(co), o r S7(o>),

(i - i , l ) n

Si(u)

39

b e t h e s e t o f I s u c h t h a t (t, / + é ) n Si(w)

0,

^ 0 , f o r e v e r y e > 0 . N o t e t h a t S ^ w ) = S ¿ ( c o ) u S + ( c o ) u S7(u).

or For

g i v e n t, a n d i„, i t f o l l o w s f r o m t h e o r e m 3 t h a t t h e f o l l o w i n g s e t s h a v e t h e s a m e p r o b a b i l i t y , under a n y initial distribution. (6.15)

{ío: U 6 Sl(u)

n S~ (co) } , {co: U G Sfr(u)

u S ~ ( c o ) } , {co: x(U,

co) = i ) .

T h e r e f o r e w e h a v e , o m i t t i n g a p r o o f t h a t t h e co-sets b e l o w b e l o n g t o (6.16)

l i m P f - S i ( c o ) n (U-e,U) «4-0 *

^0;

S< (co) n (/,, *, + « ) "

0; 1 ÚVÚ

= l i m P { 5 , (co) n ( f c - e , i , + e ) e 4- 0 *

1 g r â

—

=i} =i}

¿¡v-i) •

i V } á P { A ; * [ a ( t o ) + i , , co] = i„;

a ( w ) + U °°, (6.18)

l i m P { A ; 5 ¿ (co) f l [ a ( c o ) B C x the element y becomes y C x and the verification of y becomes the verification of y C x, that is, the mapping B C x —> (B C x) C (y C x). The resultant of the two verifications maps an element z into (z C x) C (y C x). On the other hand the resultant of the two verifications is equivalent to a verification of x y and hence z maps into z C (x y). Thus we must have the equation (3.1)

(zCx)C

(yCx)

= zC ( x ^ y ) .

The resultant of the two mappings is the mapping B C (x y). We shall require the conditional to be such that the sentences x and y C x determine the sentence y /n X. Hence ii y C x — z C x then we must have y x = z x. Finally, if x, y are elements of B and i ^ O then we require the existence of an element z such that z C x = y. This existence requirement is not essential for the present discussion but it is harmless and does result in a somewhat more elegant set of properties of the conditional. The following are the postulates for the conditional. P3.1. x, y 6 B and x ^ 0 => y C x € B. P3.2. ^ 0 = > I C I = 1 . P3.3. {~y) C x = C x) if x ^ 0. P3.4. (y C x) (z C x) = (y z) C x if z ^ 0. P3.5. z C (* / s y) = (z C x) C (y C x) if x ^ y ^ 0. P3.6. y C x = z C x and XF£0=>y^-x = z^-x. P3.7. x, y € B and x ^ 0 => there exists z such that z C x = y. We can interpret y C x as the sentence "y is implied by x" or as "x implies y." However, y C x is not equivalent either to material or strict implication. In fact the operator C cannot be defined in terms of the Boolean operators. A Boolean algebra which contains an operator C satisfying the above seven postulates is said to be implicative. Henceforth whenever we write an expression such as y C x it will be understood that x 0. The following five theorems are easily proved.

PROBABILITIES AND

PREDICTIONS

43

T3.1. ( j C i ) v ( 2 C i ) = ( ) i v z ) C x. T3.2. (yC x)+(zC x) = (y+z) C x. T3.3. (y C x) = (x y) C x. T3.4. 1 C z = 1. T3.5. xCi = x. For * C 1 = x C (1 /N 1) = (* C 1) C (1 C 1) = (x C 1) C 1 and hence x = x C 1. T h e following definition introduces a new operator denoted b y X . D3.1. (a) z C x = y and z^x=zz = x X y when x ^ 0. (b) x X y = 0 when x = 0. T3.6. x X y is unique. T3.7. x X ( y C x ) = x^y. Theorem 3.7 enables us to interpret X as the operator by means of which we can combine the sentences x and y C x to produce the sentence x ^ y. T3.8. xy = z. For letu C x = y, v C x = z. Then x^u = x X y = x X z = x^v and y = u C x = (IAS)CI = (x ^ v) C

x = v C x = z.

T3.10. xX(y + z) = ( x X y ) + (xX z). T 3 . l l . xX(y-^z) = ( x X y ) ^ ( x X z). T3.12. * X 0 = 0. T3.13. j X 1 = 1 X J ; = J;. T h e equations of T3.13 are obvious when x = 0. If x ^ 0 then i X 1 = 1 X (« C j ) = JAI = x. Moreover 1 X x — z where z = z C 1 = x. T3.14. x X y = 0=>x = 0ory = 0. T3.15. (x C y) C z = x C (y X z). For let u C y = z. T h e n u ^ y = y X z and x C (y X z) = x C (u y) = (x C y) C (u C y) = (x C y) C z. T3.16. x X (y X z) = (x X y) X For let u C (x X y) = z = (u C (u C x) y = y X z. Hence a; X (x X y) = * ^ « (x X y) = u ^

z. x) C y. T h e n NA(IXJ) = (x X y) X z (y X z) = x X ((« C x) s\ y) = ( I X ( « C I (* X y) = (* X y) X z.

and ; ) ) A

4. The ring-like character of implicative Boolean algebra W e shall give an alternative set of postulates for implicative Boolean algebra using only the operators + and X . I n terms of these two operators we shall then define the remaining Boolean operators and the operator C . These postulates will display the ringlike character of implicative Boolean algebra. P4.1. B is an Abelian group with respect to 0. P4.2. x £ B=>x + x = 0. P4.3. B is a semigroup with respect to X , 1. P4.4. x X y = x X z and x ^ 0 => y = z. P4.5. xX(y + z) = ( x X y ) + (xX z). P4.6. 0 X » = 0. P4.7. x X y = 1=> y = 1. P4.8. x,y £ B=> there exist x', y',z£ yX( 1 + *')•

B such that x X y' = y X x', (1 + x) X z =

THIRD BERKELEY SYMPOSIUM: COPELAND

44

P4.9. y, x', y', z, x", y" £ B and x ^ 0 and a; X / = y X x' and (1 + *) X z = y X (1 + x') and * X y" = y X x" => there exists w such that y" = y' Xu. These postulates can be satisfied by a system in which 1 = 0, but in this case all of the elements are 0. In the following theorems it will be assumed that this trivial case is excluded. In P4.3 it is understood that 1 acts as a unit when operating on the right. T 4 . 1 . 1 X ) C = xfor 1 X x = 1 X (1 X *) and hence x= T 4 . 2 . x X y = i=>x = y = i. T 4 . 3 . x, y £ B => there exist unique x', y' such that

l X x .

(a)

X X y' = y X x'. (b) x X y" = y X x" => there exist u, v such that x" = x' X u, y" = y' X

In the proof we consider separately three different cases. i. x = y = o =>

= y = 1.

II. x = 0, y ^ 0 => x' = 0, y = 1. x ^ 0, y = 0 => x' = 1, /

= 0.

III. x ^ 0, y ^ 0. The existence of x', y' follows from P4.8, P4.9. To prove the uniqueness suppose that there exist a pair x', y' and another pair x", y" both satisfying (a) and (b). Then there exist u, u' such that x" = x' X u, x' = x" X u' = x' X (u X u') = x' X 1. Then = 0 => x" = 0 = x' and x' ^ 0 => u X u' = 1 =» u = u' = 1 => x" = x'. Similarly y

=

y.

We have the following definitions. D 4 . 1 . xs^y = xXy' = yXx' where x', y' are defined as in T 4 . 3 . D 4 . 2 . x C y — x' if y 0, yQx — y'if where x', y' are defined as in T 4 . 3 . T4.4. x /n x = x. T4.S. X Ss y = y /N x. T4.6. (x ^ y) z = x / s (y z).

For let (a) x ^ y = xXy' = yXx', (b) (x ^ y ) ^z = xX y' X z' = y X x' X z' = zX u, ( c ) y A i = j r X 2 " = ! X y", (d) x / s (y / s z) = x X v = y X z" X x" = z X y" X x". Then (b), (c) => there exists x'" such that u = y" X x"'. Hence (e) x X y ' X z ' = zX

y" X

x'".

But (d), (e) => there exists s such that x"' = x" X s- Hence (f) ( i A j ) A j = ( i a (y a z)) X s. But x (y ^ z) = (y ^ z) ^x and (x /n y) z = z /n (X y). Therefore, there must exist t such that (g) x (y y) z) X t. z) = ((x Theorem 4.6 is then a simple consequence of (f), (g). T4.7. i X ( j A ! ) = ( i X y ) A ( i X z). For if x = 0 then x X (y ^ z) = 0 = (x X y) ^ (x X z). If x ^ 0 let y ^ z = y X z' = zXy'. Then x X (y ^ z) = x X y X z' = x X z X y'. Moreover x X y X z" = x X z X y" => y X z" = z X y" => there exist u, v such that y" = y' X u, z" = z' X v. Therefore x X (y z) = (x X y) ^ (x X z). D4.3. z v y = 1 + (1 + *) ^ (1 + y). T4.8. x v y = x + y + (x / s y).

PROBABILITIES AND PREDICTIONS

45

T4.9. a ; X ( ) v z ) = ( i X y ) v ( i X z). F o r x X ( y v 2 ) = i ( X H 2 + ( y A 2 ) ) = ( i X j ) + ( i X z ) + ((a;Xy)A(j(X2)) = (xXy)sy(xX z). T4.10. x /n (y s / z) = (z y) N/ (a; 2). For j A ( j i v 2 ) = I A ( l - f ( l - ( - j ) A ( l - f 2)) = I - f ( j a ( 1 - ) - j ) a ( 1 - ( - 2 ) ) = A A y) + 2) + ( i A y a 2) = (x y^y) ^y (x z). T4.ll. a; + y = (x ^ (1 + y)) n/ (y (1 + *))• T4.12. a; ^ (y + z) = (z y) + (as ^ z). For xs^(y+z) = a; ((y (1 + z)) (z (1 + y))) = (a; y (1 + z)) (x z + y)) = (x^y sk (1 + (x ssz))) ss (1 + (x^y))) = ( i A y ) + (xx = 0 orx= 1. With the aid of these theorems it is not difficult to verify that a system B satisfying the above nine postulates is a Boolean algebra provided v are defined by D4.1, D4.3 and is defined to be 1 + x. Moreover, the operator C defined by D4.2 satisfies postulates P3.1 to P3.7. From T4.13, T4.14 it follows that a nontrivial implicative Boolean algebra is atomless. In fact every element is infinitely divisible. Since there exist atomic Boolean algebras, the operator C cannot be defined in terms of the Boolean operators. On the other hand, it is known that an arbitrary Boolean algebra can be imbedded in an implicative Boolean algebra and hence the introduction of the operator C produces no essential restriction. In a paper as yet unpublished Professor Richard Biichi shows how to construct a natural model for every implicative Boolean algebra.

5. Probabilities We shall introduce the following postulates for probabilities: P5.1. x £ B => p(x) is a nonnegative real number. P5.2. x /n y = 0 => p(x + y) = p(x) + p(y). P5.3. p{ 1) 0. P5.4. p{x X y) = p(x)p{y). The following theorems are easily proved. T5.1. f ( l ) = 1. T5.2. 0 ^ p{x) g 1. T5.3. p{x v y) = p{x) + p(y) - p(x ^ y). T5.4. p(x ^ y ) = p{x) p{y C x) = p(y) p(x C y). For p(x ^ y) = p[x X (y C x)] = p(x) p{y C x). Independence is defined as follows. D5.1. xh « 2 , ' ' ' , xn are independent provided (5.1)

p(x1

oci /n • • •

xn) = p(x1 X xi X • • • X x„)

and provided a similar condition holds for every subset of these elements.

6. Development of the formalization We have seen that the verification of an element x is the mapping B—*BCx. The verification of x followed by the verification of y (where both x and y are regarded as elements of B) consists in the two mappings (6.1)

B^BC

x-+(BC

x) C ( y C *) = BC

(«Ay).

46

THIRD BERKELEY SYMPOSIUM: COPELAND

The verification of y followed by the verification of x consists in (6.2)

B-+B

C y->(B

C y) C (* C y) = B C ( x ^ y ) .

Hence verification is commutative. Clearly it is also associative. The verification B—>B C x maps y into yCx. IiyCx=l and the verification of x is followed by the verification of y then the second verification is the identity mapping and hence the verification of x is equivalent to the verification of x followed by the verification of y. We then say that the verification of x produces the verification of y. For the verification of an element x to produce the verification of an element y it is necessary and sufficient that x ^ y = x and also necessary and sufficient that x strictly imply y. The verification of an element x is said to produce an observation on an element y if it produces the verification either of y or of Suppose that B —> B C a produces an observation on x and an observation on y. Then this mapping produces a verification of x ^ y if it produces a verification of x and of y. Otherwise it produces a verification of y). Furthermore, the mapping produces a verification of x -v y if it produces a verification of x or of y or of both. Otherwise it produces a verification of v y). However, it is possible to produce a verification of x v y without producing an observation either on x or on y and it is possible to produce a verification of ~ ( x y) without producing an observation either on x or on y. If a mapping B—>BCa produces a verification o f i v y but does not produce observations on both x and y then we can follow the mapping by another mapping (6.3)

BC

a-+(BC

a) C b = BC

(aXb)

which produces observations on both x and y. Since the first mapping carries x y into 1 the mapping B—>BC (aX b) must carry x v y into 1 and hence must produce a verification of x or of y or of both. A verification of y) which does not produce observations on both x and y can be similarly treated. An observation on y C x shall mean a verification of x followed by an observation on y. The verification of x maps y into y C x and the observation on y then maps y C x into 1 or 0. If y C x maps into 1 it is said to be verified and if it maps into 0 then C x) is verified. No mapping which produces a verification of shall be regarded as producing an observation on y C x. If a mapping B —> B C a produces a verification of x and an observation on y then it can be regarded as the resultant of the two mappings (6.4)

B-^BC

x-+(B

C x) C {a C x) = B C

(»AJ)

=

B C a.

The second mapping carries y C x into 1 or 0. Next consider a mapping B-+ B C a which produces observations on both x and y. Then x and y map into x C a and y C a. Consider the element (6.5)

(y C a) C (x C a) = y C (* ^

a).

If x is verified by the mapping then y C (x ^ a) = y C a and y or is verified according as y C (x a) is 1 or 0. If is verified then y C (x a) is undefined. Thus we obtain a verification of y C x or verification of C x) or no observation on y C x according as (y C a) C (x C a) is 1 or 0 or undefined. It would have been much more satisfying if we could have characterized observations on y C a; in terms of the possible values of the element (6.6)

(yCx)Ca

= yC

(xX

a),

PROBABILITIES AND PREDICTIONS

47

but unfortunately such a characterization could not give us an accurate picture of the way in which observations o n y C i are actually made. It is not in general true that y C (x X a) is equal to y C (x a). We consider next the formalization of predictions. We start by assigning probabilities to the elements of B in accordance with P5.1 to P5.4. Then we choose a number X such that 0 < X < 1. We predict every element x of B for which X ^ p(x). Thus the assignment of probabilities and the selection of X produce the formalization of predictions. The usual practice is to choose X = .95. It remains to decide what can be said of those elements x for which p(x) < X. We concern ourselves with success ratios of finite sets of such elements rather than with the individual elements and hence we are led to the problem of formalizing the concept of success ratio. If we are given a set of elements x\, x2, • • •, xn then we can form the element (6.7)

yr=

U 0, D'): (a) D is the union of a monotone sequence of regular sets; (b) RS4 is satisfied for D (even if D is not regular). In the following, we shall use the PWB method to solve, in its usual generalization, the first boundary value problem on R. The substitution of R for a general open subset D

56

THIRD BERKELEY SYMPOSIUM: DOOB

of R is no real restriction, because such a set D itself can be taken as the space of the discussion, if we define the regular sets of this space as the regular sets of R whose closures are subsets of D. In the following discussion, we suppose throughout that M(i?, R') is satisfied. L e t / b e any function, not necessarily finite valued, defined on R'. The lower [upper] PWB class of functions f o r / is defined, following Brelot [1], as the class of functions on R, containing, together with the function — [uf < + ], with solution lim «„, where the

71—> co

n - * 00

latter limit exists uniformly on every compact subset of R. Properties R F 1 , R F 2 , R F 3 are easily verified. T o prove R F 4 we use a sequence {u„, n ^ 1 j of subregular functions, in the lower class for f , converging monotonely to the solution u for/. Let gn be the upper limiting function of u„ on R'. Then gn is upper semicontinuous, and we can take/i = lim gn. The function /2 can be defined similarly. Final71—> 00 ly, to prove R F 5 , we note first that, according to our analysis of lower and upper functions, U/ and uf are both regular, under the stated hypotheses, and go on by paraphrasing the proof of a special case (Brelot and Choquet [3]) in which the result is known. We treat only the monotone nonincreasing case. Suppose that [/„, n 1 j is as described in R F 5 , and that the (monotone) sequence {un, n ^ 1} has the limit u. Then (3.2)

u

n

tu^u'^uf.

We show that u/ 3: u. T o show this, let zo be any point of R, let e be a positive number, and choose vj as a lower P W B function for /,• — 1 and to satisfy (3.3)

Vj (Zo) ^ Uj (zo) - Uj-1 (zo) — ^ . «

where we have defined /o = 0, Mo = 0. Then ^ ^ Vj is a lower P W B function for /„, 1 and n

(3.4)

^

1

Vj(Zo)

>

Mn(Zo)

•

B y hypothesis, uj> — °°, so there is a lower P W B function v, not the function — 00, for/. Then v is also a lower P W B function for/„, so that (3.5)

max [ v, ^

DyJ

5«

THIRD BERKELEY SYMPOSIUM: DOOB

is a lower PWB function for /„. But then the function CO

(3.6)

max

L

t.^Dy i

is a lower PWB function f o r / n , for every n, so that this maximum is a lower PWB function f o r / . Hence CD

(3.7)

Uf (Zp) ^ ^

Vj (zp) ^ w (Zp) — e . i

Thus U/(ZQ) ^ m(z0), and it follows that % ^ a on J?. Since, as we have already noted, uf ^ u on i?, it follows that Uf — uf — u on i?, so that / is PWB resolutive. Finally, the sequence {u„, n S; 1J converges uniformly on every compact subset of R because (Dini's theorem) it is a monotone sequence of continuous functions with a continuous limit. In the classical cases, it is proved that every bounded continuous boundary function is PWB resolutive. In these cases R u R' is a complete (and usually even compact) metric space, and it will then follow from the properties listed above that, using the Riesz representation theorem or the Daniell integral definition, there is a transition measure {n(z, R, •), z f R\ such that the function / on R' is PWB resolutive if, and only if, extending our previous notation in the obvious way, / 6 L(R), and such that, if / is PWB resolutive, the corresponding solution is the average (3.8)

« (z) = f f U ) i i { s , R , d f t . J B'

The key problem in such an analysis is to prove that the bounded continuous functions on R' are PWB resolutive, and the proof of the assertion has required specific facts on the character of regular functions and the properties of boundaries. We shall approach the analysis of the characteristics of the PWB resolutive class of boundary functions entirely differently in this paper. In our discussion of the first boundary value problem, R can be replaced by any of its open subsets D, under the hypothesis M(D, D'). In particular, if D is regular, M(Z?, D') is automatically satisfied, and every continuous boundary function / is resolutive. In fact, in view of RS2, the regular average of / on D' relative to D is in both lower and upper PWB classes f o r / , and is therefore the PWB solution f o r / . The full details of the classical case, as stated in the preceding paragraph, are valid in this case, and in fact the definition of a regular set was made with this possibility in mind. Going back to the general case, we shall call R weakly PWB resolutive if M(i?, R') is satisfied and if there is a transition measure {n(z, R, •), z 6 R} such that, with the obvious extension of our customary notation, whenever / £ L(R), the regular average of / on R' relative to R, as defined by this transition measure, is regular on R, and such that, if / is PWB resolutive, t h e n / £ L(R) and / has as solution the regular average just described. If, in addition, every function in the class L(R) is PWB resolutive, R will be called PWB resolutive. If J? is PWB resolutive, and if, in addition, the domain of n{z,R, •) includes all the Borel subsets of R', for every point zoiR,R will be called strongly PWB resolutive. We have already remarked that, in the classical cases, all open sets are strongly PWB resolutive, and that, in our case, a regular set is always strongly PWB resolutive. We shall prove below that, under our usual hypothesis M(2?, R') as slightly strengthened

FIRST BOUNDARY VALUE PROBLEM

59

below to M'(R, R'), R is always weakly PWB resolutive, and that, if / € L(R), the corresponding regular average is the solution of the first boundary value problem as obtained by a natural extension of the PWB method. We shall now extend slightly the validity of a defining inequality of subregular functions. Suppose that D is an open set, whose closure is a compact subset of R, and suppose that u is a function defined and subregular on a set including the closure of A Let w on D' define the boundary function / . Then, if D is regular, and if z 6 A u(z) is less than or equal to the regular average of / relative to D, at z. This inequality remains true if M ( A D') is satisfied and if D is merely strongly PWB resolutive, where the transition measure determining the regular average is that involved in the definition of a strongly PWB resolutive set. To see this, suppose first that u is bounded on the closure of D. Then / is PWB resolutive, and u on D is a lower PWB function for / . Hence, if / has PWB solution u', u ^ u! on D, that is, u on D is at most the regular average of / on D' relative to D. In the general case, u is bounded from above on the closure of D, and the result to be proved is obtained by applying the result in the bounded case to the function max [u, n], and letting n —> — °°. It is now not difficult to prove that theorem 2.3 of section 2 is valid for A and A not necessarily regular, but satisfying M ( A , D[), M ( A , D2) and strongly PWB resolutive, if we suppose that A , A have compact closures, with A u D[ c A - In fact then, since u is subregular on a set including the closure of A , (3.9)

f

u{^)n{z,

A , dfi) ^ f

m(z, A , d f i ) f

« ( f i ) j » ( f i . D2, u monotonely, so that u is regular on an open subset of R if it is finite valued on a dense subset of this set. This remark shows, for example, that, if every bounded Borel measurable function on D' is PWB resolutive, then D is strongly PWB resolutive. In fact, each PWB solution can be expressed as the

6o

THIRD BERKELEY SYMPOSIUM: DOOB

regular average of a measure family satisfying TM1, TM2, TM3, using the Daniell integral definition, and then TM4 is automatically satisfied, so that the measure family is a transition measure.

4. The classes H, D We suppose in this section that R is strongly PWB resolutive from below, by which we mean that R can be expressed as the union of a sequence {i?„, n 1} of strongly PWB resolutive sets, with compact closures, for which Rn u R'n c i?n+1. Let z be a point of R, and let u be a function defined and subregular on R. We shall denote by H(R) the class of functions u with the property that the regular average of \u\ on R'n at z, relative to Rn, and considering only values of n so large that z £ Rn, defines a bounded sequence of numbers, for every point z of R. If u is nonnegative, these regular averages determine a monotone sequence, so that the boundedness is obviously independent of the choice of the Rn sequence. In the general case, it is easily seen that the boundedness is equivalent to the same condition for the nonnegative subregular function max [u, 0], so that u € H(R) if and only if max [u, 0] 6 H(R), and again the choice of the Rn sequence is irrelevant. We denote by D(R) the class of functions u, regular on R, such that, for each point z of R, the sequence of functions obtained by considering u on R'n for n so large, n 2: Nz, that the specified point z is in Rn, is uniformly integrable relative to the corresponding sequence {¿i(z, Rn, •)» n = Nz}, that is, that (4.1)

lim

„—»CO

f

J

| « ( f ) \n(z,Rn,

d{)

=0

uniformly for n ^ Nz. Then clearly D{R) c H(R). In the following we shall omit the set R from the notation here if there is no possible ambiguity. We shall use the following lemma to prove that the class D is independent of the choice of Rn sequence. LEMMA 4.1. For each point t in the index set T, let ¡xt be a totally finite measure on the space Xt. Let {gt, t € T} be a family of measurable functions, where gt is defined on Xt. If there is a positive function defined and monotone nondecreasing on [0, ), with lim £= (4.2)

Í-^QO

, and such that sup Jf °°, the integral on the left in (4.8) approaches zero uniformly (as n varies), for each z, because the bracketed difference on the right can be made arbitrarily small by proper choices of u\, «2, whereupon the last term goes to zero with I/a. The defining conditions for the classes H and D, as originally stated, are conditions depending on a point z of R, which are to hold for all z. In some applications, for example if R is an open set of a finite dimensional Euclidean space, or if R is a Riemann surface, and if regularity means harmonicity, the conditions are valid for all points z if they are valid for one. It is easily seen to be sufficient, for example, if there is a valid Harnack theorem, stating that whenever u is positive and regular on R, M(ZI)/M(Z2) is bounded from above, for z\ and Z2 on any specified compact subset of R, by a constant depending only on the compact set, but not on u. In this case a function u is in the class D if and only if there is a function \j/, positive, monotone nondecreasing and convex on [0, 1. Using probability language, we can now define the trajectories as follows. We choose zN at random on R'N in accordance with the probability distribution /x(z A r_i, RN, •); we then choose zN+\ at random on R'N+I, in accordance with the probability distribution ¡x(zN, RN+1, •), and so on, obtaining a random walk starting at Zo and proceeding from one boundary of a set in the nested sequence of sets to the next boundary. In more formal language, we define a stochastic process on a measure space fl(zo), with probability measure P. The integral of a random variable (function) x on i2(zo) is denoted by E{x). In particular P{O(z0)| = 1. The range of the random variable z„ is a subset of R'n, for n^ N, and z„ is measurable in the sense that the inverse image under z„ of a Borel subset of R'n is a measurable fl(z0) set. The z„ process is to be a Markov process, so that only the initial value and transition probabilities need be specified to ensure the existence of the process and to determine the joint distributions of its random variables. We have already prescribed that z0, • • •, zjV-i be identical, and have defined the transition probability measures otherwise by setting, in the usual notation, aside from the ambiguities inherent in conditional probabilities, (5.1)

P { zn+i (a>) 6 A \ zn} = n {zn, Rn+i, A) ,

n ^ N - 1.

It is of fundamental importance that, in view of the fact that Rn, A) is a regular function on Rn, and that Rm U R'm c Rn when m < n, we have (5.2) P{z„(co)6 A\zm} =n{zm,Rn, A), N-l£m — 0 0 , the w(zn) process is a semimartingale, and in fact is a semimartingale relative to the family of Borel fields { n S; 0}, where is the smallest Borel field of Q(z) sets relative to which z0, * y Zn «ire measurable. If u(za) = — 00, the parameter value 0 must be omitted from this argument. If u is regular, there is equality in (5.3), and the u(zn) process is therefore a martingale with respect to the same family of Borel fields. The following theorem is our generalization of Fatou's boundary value theorem. Note that no condition whatever has been imposed on R'. THEOREM 5.2. If R is strongly PWB resolutive from below, if u is subregular on R, and if u € H, then u has a finite limit among almost every co trajectory of any trajectory process from a point of R, and the expectation of the limit is finite. If u (E Hi then E{ |M(Z„) | j is bounded independently of n ^ 1 in the preceding theorem, because this expectation is the regular average of \u\ on R'n relative to Rn, at zo. The convergence result is now simply an application of a standard semimartingale convergence theorem to the semimartingale obtained in the preceding theorem, and, in view of Fatou's lemma .on integration to the limit, the expectation of the limit is finite. In the following, we shall hold the nested sequence {i?n, n S: 1} fast, but allow zo to vary, defining z„ as before. The limit in theorem 5.2 is a random variable x(zo), depending on Zo and in fact defined on O(zo). If u is subregular on R, and if u is bounded from above, u 6 H. More generally, if u is bounded from above by a function v £ D, then u (E H again, because H is a linear class, and v £ D c H, u — v £ H. Under such an added restriction on u, theorem 5.2 can be strengthened as follows. THEOREM 5.3. Let R be strongly PWB resolutive from below, and let u be subregular on R, and bounded from above by a function v G D. Then if the limit of u along w trajectories from z defines the random variable x(z), E{x{z)\ is finite, and (5.4)

u{z)). In particular, i j j is PWB resolutive, with solution u, then, jor almost all 03, j is constant on l?(co), and u has this constant as a limit on the co trajectory. Thus, according to this theorem, a P W B solution takes on the assigned boundary function value as a limit along our trajectories even if the trajectories do not approach individual points of R'. I n fact, as will be seen from the proof, the theorem remains t r u e if the hypothesis t h a t almost no set R(w) is e m p t y is deleted from M'(R, R'). T h e assertion of the theorem is of course not very useful for a point w for whichi?(co) is e m p t y ! If v is a lower P W B function f o r / , v is subregular and bounded from above (excluding the function — so t h a t v € H, and therefore v has a finite limit y(o>) along almost every co trajectory from z. B y definition of lower P W B functions, / y(a>) on 2?(co). If v is an upper P W B function f o r / , — v is a lower P W B function for —/, and we can apply the result just proved to derive the corresponding result. Finally, i f / is P W B resolutive,

68

THIRD B E R K E L E Y SYMPOSIUM: DOOB

with solution u, we have seen in section 4 that u £ D, and hence u has a finite limit z(co) on almost every a> trajectory from z0. Let m„ [i>„] be a function in the lower [upper] PWB class for / , with (6.1)

Mi = «2 =

'' ,

lim m„ = u [ »i 3: n—>co

^ •'' , lim n—>oo

»„=»].

We can define un, for example, as max u'j, where \u'j,j 3: 1} is a sequence of lower PWB j^n functions for / , whose upper limit at each point of a dense denumerable set on R is the value of u at the point. Let un [i>„] have the limit xn(co) [y„(co)] on an o> trajectory. Then (6.2)

xn^x^yn

(the second inequality is true with probability 1). Applying theorem 5.3, we find that (6.3)

un{za)^E{xn}

oo n—»CD

Now if co is chosen so that the limits x„(o}), yn(u>) exist for all n, it follows from the first part of the theorem that (6.5)

xn(co)^

/ ( f ) ^ y,(co)

if

t e n (a),

n ^ l .

Hence, in view of the preceding equation, we find that x(co) = / ( f ) for f G i?(co), for almost all co, as was to be proved. This theorem completes the PWB theory in a satisfactory way, by showing that, in an appropriate limit sense, a PWB resolutive boundary function is really the boundary function of its PWB solution. Without such a result, these solutions are only linked to the specified boundary functions which determine them by the PWB method itself, and this method does not give a very satisfactory connection in intuitive terms. In particular, if R is itself a regular set of a larger space in which this theory is discussed, and in this case we shall see below that almost every set i?(co) contains exactly one point, the given transition measure {n(z, R, •), z 6 J?} defines, for each Borel subset A of R', a PWB solution R, A), and theorem 6.1 thus gives us information on the limiting behavior of ¡i(z, R, A) for z near R'. We shall now extend the notion of a PWB solution by enlarging the lower and upper PWB classes for a given boundary function. The class of PWB solutions determined by the PWB resolutive boundary functions is rather unnatural. In fact this class of PWB resolutive boundary functions is clumsy, because, for example, it is not known that max [/, 0] is PWB resolutive whenever / is. This difficulty will be overcome by our extension, which will make it possible to prove that!?' is weakly PWB resolutive under the hypothesis M'(i?, R')Suppose then that M'(i?, R') is satisfied. We define the stochastic lower [upper] PWB class of functions corresponding to an arbitrary boundary function/in the way suggested by theorem 6.1. That is, the lower [upper] SPWB class consists, in addition to the function which is identically — °° [ + 0 0 ] , of all subregular [superregular] functions on R, bounded from above [below], and having the following additional property. If the function is in the lower [upper] SPWB class it has a limit y(co) ^ / ( f ) [y(co) ^ / ( f ) ] for

FIRST BOUNDARY VALUE PROBLEM

69

f £ R(w), for almost all to. According to theorem 6.1, the SPWB classes for / include the corresponding PWB classes f o r / . The existence of the limit y(oo) for almost all o> is assured by theorem 5.2. The lower [upper] SPWB solution is the supremum [infimum] of the functions in the lower [upper] SPWB class, and is regular, if finite on a dense set. Moreover, it is in the lower [upper] SPWB class, if bounded from above [below], and is greater [less] than or equal to the lower [upper] PWB solution. Finally,/ will be called SPWB resolutive if the lower and upper SPWB solutions are regular and equal, and the common solution will then be called the SPWB solution for /. This solution is in both lower and upper SPWB classes for / , if it is bounded. We now see that, always under M'(R, R'), i f / is PWB resolutive, it is SPWB resolutive. Theorem 4.1 is applicable, and shows that every SPWB solution is in the class D. Theorem 6.1 and its proof remain valid for the SPWB method. The properties RF1 to RF5 of the class of PWB resolutive boundary functions remain valid for the class of SPWB resolutive boundary functions, and in this version will be denoted by SRF1 to SRF5. In addition, however, the new class has the following property. SRF6. If f is SPWB resolutive, and if $ is a convex monotone nondecreasing function, bounded from below, defined on{— °°, ), then 4>(f) is SPWB resolutive if its upper SPWB solution is finite. The most important application of this property, whose validity we shall prove in the next paragraph, is to prove that, if / and g are SPWB resolutive, then max [/, g] is also. In fact this assertion for g = 0 follows directly from SRF6, and follows in general from the equality (6.6)

'

max [ / , g] = m a x [ / - g, 0] + g .

PROOF OF S R F 6 . L e t u b e t h e S P W B s o l u t i o n f o r / . If u0 is a f u n c t i o n in t h e S P W B

lower class for/, .

3) Let g) be a second space, Q a Borel field of subsets mC let p(x; co) be a probability measure on Q corresponding to every» 6 X , and let Y(x) be a random element taking its values on g) obeying the law p(x; co). I assume that the different Y(x) for different x are mutually independent. Let e 6 £ with m(e) < + , let Xu X2, • • •, Xj, • • • be the elements of F belonging to e. Then the Y(Xj) are Poisson distributed on §) (with respect to Q) and the mathematical expectation of the number of the F(X,) belonging to co is (1.2) This paper was prepared with partial support of the Office of Naval Research.

8i

82

THIRD BERKELEY SYMPOSIUM: FORTET

This last property is useful in some physical problems (in such problems, usually y f a n d £) are the time axis, or some Euclidean spaces).

2. Random distributions (by Gelfand) We saw in section 1 t h a t it is possible to define, and to handle directly, Poisson distributions. I t m a y be expected t h a t the same is true for all random distributions which are almost surely (a.s.) purely discrete distributions (these distributions, in the case of distributions on the straight line, were called "point processes" by H . Wold). B u t for some problems it m a y be easier to use an indirect way. On the other hand, one encounters, in m a n y questions, distributions which are not purely discrete distributions, and in such a case it seems t h a t we have no way except an indirect one. Roughly speaking, we can consider a distribution as a linear functional on some vectorial function space 2b', this must be the starting point. As an example of what can be done in this direction, I take a recent paper by Gelfand [2], which is concerned with distributions on the time axis, — oo < t < + c°. As a space 2b, he takes L. Schwartz's 2b space of the f u n c t i o n s / ( / ) which are indefinitely differentiable and equal to 0 outside a compact set with the usual pseudotopology (see p. 24 and p. 66 in [3]). Any continuous linear functional F on this space 2b is an L . Schwartz distribution. T h e space of these distributions is the dual space 2b' of 2bLet us consider also two other spaces: the space 2 b * consisting of all the linear funct i o n a l on 2b, which are continuous or not, and the space £ consisting of all the funct i o n a l on 2b, which are linear or not, continuous or not. We then have 2b' C 2b* CIf we think of a random element F taking its values in ¿J, we can consider the family of the random variables X j = (F, f ), where (F, / ) is the number obtained b y applying to / € 2b the functional F f £ ; and we can say t h a t F, as a random element, [fx, • • •, fk] have the following continuity property. (k> Continuity property or property (B). For any fixed k and fixed/x, • • •, fk, lim [/i n ) , »—•+00 '' ">/fc(n>] = • •' ,fh] in the sense of the usual convergence of distribution functions. Under the name of "general random process" (G.R.P.), Gelfand considers random elements F in ¿J having properties (A) and (B). These properties together do not necessarily imply t h a t F takes its values in 2>', or even in 2>*, b u t it is worth noticing t h a t t h e y imply t h a t for a n y fixed k and fixed _/i, • • •, fk, if (2.4) lim /$"> = / , , ¿=1,2, •••,«, »—>+00 ' ' then ••, Xf™} tends in probability to \Xh,- • Xh\. Gelfand defines as the characteristic functional, 0, we can find (i) an indexing y / x ) of y(x), (ii) a number -q = y (e, x0) > 0, and (iii) a permutation ttj, t2, • • •, tt„ of the integers 1, 2, • • •, n such that (3.2) \yj(xo) -J/T^X) I 0

( z K — e*1!' U,t

=